[Hinton06] showed that RBMs can be stacked and trained in a greedy manner
to form so-called Deep Belief Networks (DBN). DBNs are graphical models which
learn to extract a deep hierarchical representation of the training data.
They model the joint distribution between observed vector and
the hidden layers as follows:

(1)

where , is a conditional distribution
for the visible units conditioned on the hidden units of the RBM at level
, and is the visible-hidden joint
distribution in the top-level RBM. This is illustrated in the figure below.

The principle of greedy layer-wise unsupervised training can be applied to
DBNs with RBMs as the building blocks for each layer [Hinton06], [Bengio07].
The process is as follows:

1. Train the first layer as an RBM that models the raw input as its visible layer.

2. Use that first layer to obtain a representation of the input that will
be used as data for the second layer. Two common solutions exist. This
representation can be chosen as being the mean activations
or samples of .

3. Train the second layer as an RBM, taking the transformed data (samples or
mean activations) as training examples (for the visible layer of that RBM).

4. Iterate (2 and 3) for the desired number of layers, each time propagating
upward either samples or mean values.

5. Fine-tune all the parameters of this deep architecture with respect to a
proxy for the DBN log- likelihood, or with respect to a supervised training
criterion (after adding extra learning machinery to convert the learned
representation into supervised predictions, e.g. a linear classifier).

In this tutorial, we focus on fine-tuning via supervised gradient descent.
Specifically, we use a logistic regression classifier to classify the input
based on the output of the last hidden layer of the
DBN. Fine-tuning is then performed via supervised gradient descent of the
negative log-likelihood cost function. Since the supervised gradient is only
non-null for the weights and hidden layer biases of each layer (i.e. null for
the visible biases of each RBM), this procedure is equivalent to initializing
the parameters of a deep MLP with the weights and hidden layer biases obtained
with the unsupervised training strategy.

Why does such an algorithm work ? Taking as example a 2-layer DBN with hidden
layers and (with respective weight parameters
and ), [Hinton06] established
(see also Bengio09]_ for a detailed derivation) that can be rewritten as,

(2)

represents the KL divergence between
the posterior of the first RBM if it were standalone, and the
probability for the same layer but defined by the entire DBN
(i.e. taking into account the prior defined by the
top-level RBM). is the entropy of the distribution
.

It can be shown that if we initialize both hidden layers such that
, and the KL
divergence term is null. If we learn the first level RBM and then keep its
parameters fixed, optimizing Eq. (2) with respect
to can thus only increase the likelihood .

Also, notice that if we isolate the terms which depend only on , we
get:

Optimizing this with respect to amounts to training a second-stage
RBM, using the output of as the training distribution,
when is sampled from the training distribution for the first RBM.

To implement DBNs in Theano, we will use the class defined in the Restricted Boltzmann Machines (RBM)
tutorial. One can also observe that the code for the DBN is very similar with the one
for SdA, because both involve the principle of unsupervised layer-wise
pre-training followed by supervised fine-tuning as a deep MLP.
The main difference is that we use the RBM class instead of the dA
class.

We start off by defining the DBN class which will store the layers of the
MLP, along with their associated RBMs. Since we take the viewpoint of using
the RBMs to initialize an MLP, the code will reflect this by seperating as
much as possible the RBMs used to initialize the network and the MLP used for
classification.

classDBN(object):"""Deep Belief Network A deep belief network is obtained by stacking several RBMs on top of each other. The hidden layer of the RBM at layer `i` becomes the input of the RBM at layer `i+1`. The first layer RBM gets as input the input of the network, and the hidden layer of the last RBM represents the output. When used for classification, the DBN is treated as a MLP, by adding a logistic regression layer on top. """def__init__(self,numpy_rng,theano_rng=None,n_ins=784,hidden_layers_sizes=[500,500],n_outs=10):"""This class is made to support a variable number of layers. :type numpy_rng: numpy.random.RandomState :param numpy_rng: numpy random number generator used to draw initial weights :type theano_rng: theano.tensor.shared_randomstreams.RandomStreams :param theano_rng: Theano random generator; if None is given one is generated based on a seed drawn from `rng` :type n_ins: int :param n_ins: dimension of the input to the DBN :type hidden_layers_sizes: list of ints :param hidden_layers_sizes: intermediate layers size, must contain at least one value :type n_outs: int :param n_outs: dimension of the output of the network """self.sigmoid_layers=[]self.rbm_layers=[]self.params=[]self.n_layers=len(hidden_layers_sizes)assertself.n_layers>0ifnottheano_rng:theano_rng=MRG_RandomStreams(numpy_rng.randint(2**30))# allocate symbolic variables for the dataself.x=T.matrix('x')# the data is presented as rasterized imagesself.y=T.ivector('y')# the labels are presented as 1D vector# of [int] labels

self.sigmoid_layers will store the feed-forward graphs which together form
the MLP, while self.rbm_layers will store the RBMs used to pretrain each
layer of the MLP.

Next step, we construct n_layers sigmoid layers (we use the
HiddenLayer class introduced in Multilayer Perceptron, with the only modification
that we replaced the non-linearity from tanh to the logistic function
) and n_layers RBMs, where n_layers
is the depth of our model. We link the sigmoid layers such that they form an
MLP, and construct each RBM such that they share the weight matrix and the
hidden bias with its corresponding sigmoid layer.

foriinxrange(self.n_layers):# construct the sigmoidal layer# the size of the input is either the number of hidden# units of the layer below or the input size if we are on# the first layerifi==0:input_size=n_inselse:input_size=hidden_layers_sizes[i-1]# the input to this layer is either the activation of the# hidden layer below or the input of the DBN if you are on# the first layerifi==0:layer_input=self.xelse:layer_input=self.sigmoid_layers[-1].outputsigmoid_layer=HiddenLayer(rng=numpy_rng,input=layer_input,n_in=input_size,n_out=hidden_layers_sizes[i],activation=T.nnet.sigmoid)# add the layer to our list of layersself.sigmoid_layers.append(sigmoid_layer)# its arguably a philosophical question... but we are# going to only declare that the parameters of the# sigmoid_layers are parameters of the DBN. The visible# biases in the RBM are parameters of those RBMs, but not# of the DBN.self.params.extend(sigmoid_layer.params)# Construct an RBM that shared weights with this layerrbm_layer=RBM(numpy_rng=numpy_rng,theano_rng=theano_rng,input=layer_input,n_visible=input_size,n_hidden=hidden_layers_sizes[i],W=sigmoid_layer.W,hbias=sigmoid_layer.b)self.rbm_layers.append(rbm_layer)

self.logLayer=LogisticRegression(input=self.sigmoid_layers[-1].output,n_in=hidden_layers_sizes[-1],n_out=n_outs)self.params.extend(self.logLayer.params)# compute the cost for second phase of training, defined as the# negative log likelihood of the logistic regression (output) layerself.finetune_cost=self.logLayer.negative_log_likelihood(self.y)# compute the gradients with respect to the model parameters# symbolic variable that points to the number of errors made on the# minibatch given by self.x and self.yself.errors=self.logLayer.errors(self.y)

The class also provides a method which generates training functions for each
of the RBMs. They are returned as a list, where element is a
function which implements one step of training for the RBM at layer
.

defpretraining_functions(self,train_set_x,batch_size,k):'''Generates a list of functions, for performing one step of gradient descent at a given layer. The function will require as input the minibatch index, and to train an RBM you just need to iterate, calling the corresponding function on all minibatch indexes. :type train_set_x: theano.tensor.TensorType :param train_set_x: Shared var. that contains all datapoints used for training the RBM :type batch_size: int :param batch_size: size of a [mini]batch :param k: number of Gibbs steps to do in CD-k / PCD-k '''# index to a [mini]batchindex=T.lscalar('index')# index to a minibatch

In order to be able to change the learning rate during training, we associate a
Theano variable to it that has a default value.

learning_rate=T.scalar('lr')# learning rate to use# number of batchesn_batches=train_set_x.get_value(borrow=True).shape[0]/batch_size# begining of a batch, given `index`batch_begin=index*batch_size# ending of a batch given `index`batch_end=batch_begin+batch_sizepretrain_fns=[]forrbminself.rbm_layers:# get the cost and the updates list# using CD-k here (persisent=None) for training each RBM.# TODO: change cost function to reconstruction errorcost,updates=rbm.get_cost_updates(learning_rate,persistent=None,k=k)# compile the theano functionfn=theano.function(inputs=[index,theano.Param(learning_rate,default=0.1)],outputs=cost,updates=updates,givens={self.x:train_set_x[batch_begin:batch_end]})# append `fn` to the list of functionspretrain_fns.append(fn)returnpretrain_fns

Now any function pretrain_fns[i] takes as arguments index and
optionally lr – the learning rate. Note that the names of the parameters
are the names given to the Theano variables (e.g. lr) when they are
constructed and not the name of the python variables (e.g. learning_rate). Keep
this in mind when working with Theano. Optionally, if you provide k (the
number of Gibbs steps to perform in CD or PCD) this will also become an
argument of your function.

In the same fashion, the DBN class includes a method for building the
functions required for finetuning ( a train_model, a validate_model
and a test_model function).

defbuild_finetune_functions(self,datasets,batch_size,learning_rate):'''Generates a function `train` that implements one step of finetuning, a function `validate` that computes the error on a batch from the validation set, and a function `test` that computes the error on a batch from the testing set :type datasets: list of pairs of theano.tensor.TensorType :param datasets: It is a list that contain all the datasets; the has to contain three pairs, `train`, `valid`, `test` in this order, where each pair is formed of two Theano variables, one for the datapoints, the other for the labels :type batch_size: int :param batch_size: size of a minibatch :type learning_rate: float :param learning_rate: learning rate used during finetune stage '''(train_set_x,train_set_y)=datasets[0](valid_set_x,valid_set_y)=datasets[1](test_set_x,test_set_y)=datasets[2]# compute number of minibatches for training, validation and testingn_valid_batches=valid_set_x.get_value(borrow=True).shape[0]n_valid_batches/=batch_sizen_test_batches=test_set_x.get_value(borrow=True).shape[0]n_test_batches/=batch_sizeindex=T.lscalar('index')# index to a [mini]batch# compute the gradients with respect to the model parametersgparams=T.grad(self.finetune_cost,self.params)# compute list of fine-tuning updatesupdates=[]forparam,gparaminzip(self.params,gparams):updates.append((param,param-gparam*learning_rate))train_fn=theano.function(inputs=[index],outputs=self.finetune_cost,updates=updates,givens={self.x:train_set_x[index*batch_size:(index+1)*batch_size],self.y:train_set_y[index*batch_size:(index+1)*batch_size]})test_score_i=theano.function([index],self.errors,givens={self.x:test_set_x[index*batch_size:(index+1)*batch_size],self.y:test_set_y[index*batch_size:(index+1)*batch_size]})valid_score_i=theano.function([index],self.errors,givens={self.x:valid_set_x[index*batch_size:(index+1)*batch_size],self.y:valid_set_y[index*batch_size:(index+1)*batch_size]})# Create a function that scans the entire validation setdefvalid_score():return[valid_score_i(i)foriinxrange(n_valid_batches)]# Create a function that scans the entire test setdeftest_score():return[test_score_i(i)foriinxrange(n_test_batches)]returntrain_fn,valid_score,test_score

Note that the returned valid_score and test_score are not Theano
functions, but rather Python functions. These loop over the entire
validation set and the entire test set to produce a list of the losses
obtained over these sets.

numpy_rng=numpy.random.RandomState(123)print'... building the model'# construct the Deep Belief Networkdbn=DBN(numpy_rng=numpy_rng,n_ins=28*28,hidden_layers_sizes=[1000,1000,1000],n_outs=10)

There are two stages in training this network: (1) a layer-wise pre-training and
(2) a fine-tuning stage.

For the pre-training stage, we loop over all the layers of the network. For
each layer, we use the compiled theano function which determines the
input to the i-th level RBM and performs one step of CD-k within this RBM.
This function is applied to the training set for a fixed number of epochs
given by pretraining_epochs.

With the default parameters, the code runs for 100 pre-training epochs with
mini-batches of size 10. This corresponds to performing 500,000 unsupervised
parameter updates. We use an unsupervised learning rate of 0.01, with a
supervised learning rate of 0.1. The DBN itself consists of three
hidden layers with 1000 units per layer. With early-stopping, this configuration
achieved a minimal validation error of 1.27 with corresponding test
error of 1.34 after 46 supervised epochs.

On an Intel(R) Xeon(R) CPU X5560 running at 2.80GHz, using a multi-threaded MKL
library (running on 4 cores), pretraining took 615 minutes with an average of
2.05 mins/(layer * epoch). Fine-tuning took only 101 minutes or approximately
2.20 mins/epoch.

Hyper-parameters were selected by optimizing on the validation error. We tested
unsupervised learning rates in and supervised
learning rates in . We did not use any form of
regularization besides early-stopping, nor did we optimize over the number of
pretraining updates.

One way to improve the running time of your code (given that you have
sufficient memory available), is to compute the representation of the entire
dataset at layer i in a single pass, once the weights of the
-th layers have been fixed. Namely, start by training your first
layer RBM. Once it is trained, you can compute the hidden units values for
every example in the dataset and store this as a new dataset which is used to
train the 2nd layer RBM. Once you trained the RBM for layer 2, you compute, in
a similar fashion, the dataset for layer 3 and so on. This avoids calculating
the intermediate (hidden layer) representations, pretraining_epochs times
at the expense of increased memory usage.